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160                     ON  THE  COLOURS  OF  THICK   PLATES.
investigated in the first section; but on account of the interest which attaches itself to these bands, and the simplicity of their theory, a separate investigation is likewise given. The next two sections are devoted to cases of more generality, and on the whole less interest: still, a few results of some interest are obtained. The last two sections contain a closer examination of the precise
Although the present paper is a little long, the reader must not suppose that the theory of the rings and bands is anything but simple.    The  length arises partly from the detail in which the subject has been considered, partly from the generality of some of ,. fi                            the  investigations,  partly from the description  of experiments
Jj]                            which accompanies the theoretical investigations.
SECTION I.
Rings thrown on a screen "by a concave mirror consisting of a lens dimmed at the first surface, and quicksilvered (it the back. Condition of distinctness when the rings are thrown on a screen, or of fixity when they are viewed in< air. Invcstifftttiou of the phenomena observed when the luminous point is '/nouedin a direction perpendicular to the mis of the mirror.
1. Let a luminous point L be situated either in or not far out of the axis of a mirror such as that just described ; and lot, it be required to investigate the illumination, at the point M of a screen, due to two streams of light, of which one is scattered at the first surface, and then regularly reflected and refracted, and the other is regularly refracted and reflected, and then scattered in coming out, the point M being supposed to be situated not far out of the axis. Let the mirror be referred to the rectangular axes of x, y, z, the axis of z being the axis of the mirror, and UK.; origin being situated in the first or dimmed surface4. Let r be the radius of the first surface, s that of the second, t the thickness of the glass,/x its index of refraction; and suppose r and ,v positive when the concavities of both surfaces are turned in the direction of # positive. Let a, b, c be the co-ordinates of L\ (tf, //, c' those of M\ and suppose a, 6, a, and b' small compared wit-h c-} c, ?', and ,v. Let x, y, z be the co-ordinates of any point, /' on the dimmed